Section 3.4

Library of Functions;

Piecewise-defined Functions

Graph the Functions Listed in the Library of Functions

Constant Function ( )f x bx y=b

-3

-2

-1

0

1

2

3

Dom ( , )f

Ran f b

-intercept

none if 0

x

b

-intercept

y

y b

( ) is an even function.y f x b

Identity Function ( )f x xx y=x

-3

-2

-1

0

1

2

3

Dom ( , )f

Ran ( , )f

-intercept

0

x

x

-intercept

0

y

y

( ) is an odd function.y f x x

Square Function 2( )f x xx y=x2

-3

-2

-1

0

1

2

3

Dom ( , )f

Ran [0, )f

-intercept

0

x

x

-intercept

0

y

y

2( ) is an even function.y f x x

Cube Function 3( )f x xx y=x3

-3

-2

-1

0

1

2

3

Dom ( , )f

Ran ( , )f

-intercept

0

x

x

-intercept

0

y

y

3( ) is an odd function.y f x x

Square Root Function ( )f x xx y=x1/2

0

1

2

3

4

7

9

Dom [0, )f

Ran [0, )f

-intercept

0

x

x

-intercept

0

y

y

( ) is neither odd nor even.y f x x

Cube Root Function 3( )f x xx y=x1/3

-27

-8

-1

0

1

8

27

Dom ( , )f

Ran ( , )f

-intercept

0

x

x

-intercept

0

y

y

3( ) is an odd function.y f x x

Reciprocal Function ( ) 1f x xx y=1/x

4

2

1

1/2

1/3

1/4

0

Dom

( ,0) (0, )

f

Ran

( ,0) (0, )

f

-intercept

none

x

-intercept

none

y

1( ) is an odd function.y f x

x

( )f x xx y=|x|

-3

-2

-1

0

1

2

3

Absolute Value Function

Ran [0, )f

-intercept

0

x

x

-intercept

0

y

y

( ) is an even function.y f x x

Dom ( , )f

Greatest Integer Functionx y= x

-1

-0.5

0

0.5

1

1.5

2

Dom ( , )f

Ran f

-intercept

0

x

x

-intercept

0

y

y

( ) is neither odd nor even.y f x x « ®

Piecewise-defined Functions

expression 1 if condition 1

expression 2 if condition 2( )

expression n if condition

f x

n

Use when x values satisfy condition n

Use when x values satisfy condition 1

Sometimes we need more than one formula to specify a function algebraically. In this case the formula used to evaluate the function depends on the value of x.

Piecewise Defined Functions

The following is a quick example of a piecewise defined function

2

32 5.5 if 2( )

13.8 2.5 if 2

x xf x

x x

(1) 32 5.5(1)f 2(4) 13.8 2.5(4)f

= 26.5

= 53.8

Use when x values are greater than 2

Use when x values are less than or equal to 2

Notice

Example 1

2

32 5.5 if 2( )

13.8 2.5 if 2

x xf x

x x

Notice that the domain of f , in this case, is the set all real numbers. That is, Dom f = (– , )

The following is a quick example of a piecewise defined function

Example 1

The percentage p (t) of buyers of new cars who used the Internet for research or purchase since 1997 is given by the following function.† (t = 0 represents 1997).

10 15 if 0 1( )

15 10 if 1 4

t tp t

t t

Notice that the domain of p is the interval [0 , 4]. That is, Dom p = [0 , 4].

†The model is based on data through 2000. Source: J.D. Power Associates/The New York Times, January 25, 2000, p. C1

Example 2

10 15 if 0 1( )

15 10 if 1 4

t tp t

t t

This notation tells us that we use the first formula, 10t + 15, if 0 t < 1, or, t is in [0, 1)

the second formula, 15t + 10, if 1 t 4, or, t is in [1,4]

Example 2

10 15 if 0 1( )

15 10 if 1 4

t tp t

t t

Thus, for instance, p(0.5) = 10(0.5) + 15 = 20 Here we used the first formula since

0 0.5 < 1, or, equivalently, 0.5 is in [0, 1).

p(2) = 15(2) + 10 = 40 We used the second formula since 1 2 4, or equivalently, 2 is in [1,

4]. p(4.1) is undefined p (t ) is only defined if 0 t 4.

Example 2

The function f (x) is defined as

2

1 if 1 1

( ) 2 if 1

if 1

x x

f x x

x x

Example 3

(a)

(b)

Example 4: Cost of Electricity

Example 4: Cost of Electricity

Example 4: Cost of Electricity

Example 4: Cost of Electricity

Therefore the function C (x) can be written as